det(UV) = det(U)·det(V).
And also, in the case of an invertible matrix U, we have:
det(U -1) = 1/det(U) ,
because the identity matrix has determinant = 1 and U·U -1 = 1 (the identity matrix). Well, maybe one could claim that the formula for det(U -1) isn't such a big deal in the case where U is orthogonal, since we have det(U) = ±1 in this case anyway. Nonetheless, it still could be the right way to think about things when you're considering conjugate matrices.
Supplementary exercise #3
[Optional; please submit separately] Given triangle
ABC
and triangle
PQR. Assume that
there exists a nonzero
real number v such that:
|
| = v |
|
|
| = v |
|
and
|
| = v |
|.
Show that if the orthocenter of triangle
ABC
is (r,s,t)
ABC, then the orthocenter of triangle PQR is
(r,s,t)PQR. (We use
the notations ABC and PQR as part of
the superscript to indicate the triangle relative to which we are taking
the barycentric coordinates . . . )
Suggestion: Refer to the formula from Theorem 7 of Chapter 3
of the text. What power of v can be factored from
the numerators and from the denominators?
( It's useful to note here that the numerators and
denominators in the formula given in Theorem 7 are homogeneous
polynomials: this means that all of the monomials occurring in
one of these polynomials have the same total degree.
For instance,
2x3 + 3y3 + 5z3 + 11xyz is homogeneous of degree 3.
)
Note: The assignment also includes some exercises from the text.
Comments and questions top;
roberts@math.umn.edu
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